Elliptic ruled surfaces over arbitrary characteristic fields

Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a ${\mathbb{P}}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah’s classification, and what kinds of singular fibers appear.